Quadratic roots, medium range

Average Error: 43.8 → 0.2

Time: 5.1s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

C

double f(double a, double b, double c) {
        double r29353 = b;
        double r29354 = -r29353;
        double r29355 = r29353 * r29353;
        double r29356 = 4.0;
        double r29357 = a;
        double r29358 = r29356 * r29357;
        double r29359 = c;
        double r29360 = r29358 * r29359;
        double r29361 = r29355 - r29360;
        double r29362 = sqrt(r29361);
        double r29363 = r29354 + r29362;
        double r29364 = 2.0;
        double r29365 = r29364 * r29357;
        double r29366 = r29363 / r29365;
        return r29366;
}

↓

double f(double a, double b, double c) {
        double r29367 = 1.0;
        double r29368 = 2.0;
        double r29369 = r29367 / r29368;
        double r29370 = 4.0;
        double r29371 = c;
        double r29372 = r29370 * r29371;
        double r29373 = b;
        double r29374 = -r29373;
        double r29375 = r29373 * r29373;
        double r29376 = a;
        double r29377 = r29370 * r29376;
        double r29378 = r29377 * r29371;
        double r29379 = r29375 - r29378;
        double r29380 = sqrt(r29379);
        double r29381 = r29374 - r29380;
        double r29382 = r29372 / r29381;
        double r29383 = r29369 * r29382;
        return r29383;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.8

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm

3. Applied flip-+ 43.8

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

4. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
5. Using strategy rm

6. Applied *-un-lft-identity 0.4

   \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

7. Applied *-un-lft-identity 0.4

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

8. Applied times-frac 0.4

   \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

9. Applied times-frac 0.4

   \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}}\]

10. Simplified 0.4

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]

11. Simplified 0.4

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
12. Using strategy rm

13. Applied associate-/r* 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]
14. Using strategy rm

15. Applied *-un-lft-identity 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot a}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

16. Applied times-frac 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{a}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

17. Simplified 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{4} \cdot \frac{a \cdot c}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

18. Simplified 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \color{blue}{c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

19. Final simplification 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))